Introduction to Higher Order Functions Dr. Mattox Beckman - - PowerPoint PPT Presentation

First Class Anonymous Functions

Introduction to Higher Order Functions

• Dr. Mattox Beckman

University of Illinois at Urbana-Champaign Department of Computer Science

First Class Anonymous Functions

Objectives

◮ Explain the concept of fjrst class citizen. ◮ Use sectioning and lambda to defjne anonymous functions. ◮ Change the behavior and interface of a function by using another function.

First Class Anonymous Functions

First Class Functions

An entity is said to be fjrst class when it can be:

◮ Assigned to a variable, passed as a parameter, or returned as a result

Examples:

◮ APL: scalars, vectors, arrays ◮ C: scalars, pointers, structures ◮ C++: like C, but with objects ◮ Haskell, Lisp, OCaml: scalars, lists, tuples, functions

The Kind of Data a Program Manipulates Changes the Expressive Ability of a Program.

First Class Anonymous Functions

Defjning Functions the Usual Way

Some Haskell Functions

1 sqr a = a * a 2 hypotsq a b = sqr a + sqr b

Sample Run

1 sqr :: Integer -> Integer 2 sqr :: Num a => a -> a 3 hypotsq :: Num a => a -> a -> a 4 Prelude> sqr 10 5 100 6 Prelude> hypotsq 3 4 7 25

First Class Anonymous Functions

Example: Compose

Example

1 inc x = x + 1 2 double x = x * 2 3 compose f g x = f (g x)

◮ Notice the function types.

1 compose :: (t1 -> t2) -> (t -> t1) -> t -> t2 2 Prelude> :t double 3 double :: Integer -> Integer 4 Prelude> double 10 5 20 6 Prelude> compose inc double 10 7 21

f g

First Class Anonymous Functions

Example: Twice

◮ One handy function allows us to do something twice. ◮ You will see this function again!

Twice

1 twice f x = f (f x)

Here is a sample run … Prelude> :t twice twice :: (t -> t) -> t -> t Prelude> twice inc 5 7 Prelude> twice twice inc 4

First Class Anonymous Functions

Creating Functions: Lambda Form

◮ Functions do not have to have names.

1 \x -> x + 1

◮ The parts:

◮ Backslash (a.k.a. lambda) ◮ Parameter list ◮ Arrow ◮ Body of function 1 prelude> (\x -> x + 1) 41 2 42

First Class Anonymous Functions

Creating Functions: Partial Application

Standard Form vs. Anonymous Form

1 inc :: (Num t) => t -> t 2 inc a = a + 1 3 inc = \a -> a + 1 4 5 plus :: (Num t) => t -> t -> t 6 plus a b = a + b 7 plus = \a -> \b -> a + b

◮ What do you think we would get if we called plus 1 ?

First Class Anonymous Functions

Creating Functions: Partial Application

Standard Form vs. Anonymous Form

1 inc :: (Num t) => t -> t 2 inc a = a + 1 3 inc = \a -> a + 1 4 5 plus :: (Num t) => t -> t -> t 6 plus a b = a + b 7 plus = \a -> \b -> a + b

◮ What do you think we would get if we called plus 1 ?

1 inc = plus 1

First Class Anonymous Functions

η-equivalence

An Equivalence

f ≡ \x -> f x

◮ Proof, assuming f is a function…

f z ≡ (\x -> f x) z

These are Equivalent

1 plus a b = (+) a b 2 plus a = (+) a 3 plus = (+)

So are These

1 inc x = x + 1 2 inc = (+) 1 3 inc = (+1)

First Class Anonymous Functions

Curry and Uncurry

◮ Suppose you have a function tplus that takes a pair of integers and adds them.

1 tplus :: (Integer,Integer) -> Integer 2 tplus (a,b) = a + b

◮ But you really wish it took its arguments one at a time. ◮ There’s a function curry :: (a,b) -> c -> a -> b -> c that will convert it for

you! See if you can write it.